Multiplication and Addition

Oct 13, 2025 by miodeya.com 115 views

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division. In this article, we will explore the concept of synthetic division and apply it to the given division problem.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut to polynomial long division and is used to divide polynomials by factors of the form (x - a) or (x + a). The process involves writing down the coefficients of the polynomial, the root of the linear factor, and then performing a series of multiplications and additions to find the quotient and remainder.

The Division Problem

The given division problem is:

$\frac{6x^2 - x + 5}{x + 7}$

We need to express this division problem in synthetic division form.

Synthetic Division Form

Synthetic division form is a compact way of expressing the division problem. It involves writing down the coefficients of the polynomial, the root of the linear factor, and then performing a series of multiplications and additions to find the quotient and remainder.

Option A

Option A is:

-7 \longdiv {6} -1 \quad 5

This option is incorrect because the root of the linear factor is -7, but the coefficients of the polynomial are not written in the correct order.

Option B

Option B is:

-7 \longdiv {6} 1 \begin{array}{lll}{5}\end{array}

This option is also incorrect because the root of the linear factor is -7, but the coefficients of the polynomial are not written in the correct order.

Option C

Option C is:

7 \longdiv {6} -1 \quad 5

This option is correct because the root of the linear factor is -7, and the coefficients of the polynomial are written in the correct order.

Conclusion

In conclusion, the correct synthetic division form for the given division problem is:

7 \longdiv {6} -1 \quad 5

This option correctly expresses the division problem in synthetic division form.

Step-by-Step Solution

To solve this problem, we need to follow these steps:

Write down the coefficients of the polynomial: 6, -1, 5
Write down the root of the linear factor: -7
Perform a series of multiplications and additions to find the quotient and remainder.

Multiplication and Addition

To perform the multiplication and addition, we need to follow these steps:

Multiply the root of the linear factor by the first coefficient: -7 * 6 = -42
Add the result to the second coefficient: -42 + (-1) = -43
Multiply the root of the linear factor by the result: -7 * -43 = 301
Add the result to the third coefficient: 301 + 5 = 306

Quotient and Remainder

The quotient is the result of the division, and the remainder is the result of the multiplication and addition.

Quotient

The quotient is:

6x - 43

Remainder

The remainder is:

306

Final Answer

The final answer is:

7 \longdiv {6} -1 \quad 5

Q&A: Synthetic Division

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a simplified approach to polynomial division that eliminates the need for long division.

Q: What are the steps involved in synthetic division?

A: The steps involved in synthetic division are:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Perform a series of multiplications and additions to find the quotient and remainder.

Q: What is the difference between synthetic division and long division?

A: Synthetic division is a shortcut to polynomial long division. It eliminates the need for long division and is used to divide polynomials by factors of the form (x - a) or (x + a).

Q: How do I know which option is correct in synthetic division?

A: To determine which option is correct, you need to follow these steps:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Perform a series of multiplications and additions to find the quotient and remainder.

Q: What is the quotient in synthetic division?

A: The quotient is the result of the division. It is the polynomial that results from dividing the original polynomial by the linear factor.

Q: What is the remainder in synthetic division?

A: The remainder is the result of the multiplication and addition. It is the value that remains after dividing the original polynomial by the linear factor.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b)?

A: No, synthetic division can only be used to divide polynomials by linear factors of the form (x - a) or (x + a).

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0?

A: Yes, synthetic division can be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if b = 0?

A: Yes, synthetic division can be used to divide polynomials by factors of the form (x^2 + ax + b) if b = 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0 and b = 0?

A: Yes, synthetic division can be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0 and b = 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b ≠ 0?

A: No, synthetic division cannot be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b ≠ 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b = 0?

A: Yes, synthetic division can be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b = 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0 and b ≠ 0?

A: Yes, synthetic division can be used to divide polynomials by factors of the form (x^2 + ax + b) if a = 0 and b ≠ 0.

Q: Can synthetic division be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b ≠ 0?

A: No, synthetic division cannot be used to divide polynomials by factors of the form (x^2 + ax + b) if a ≠ 0 and b ≠ 0.

Conclusion

In conclusion, synthetic division is a simplified approach to polynomial division that eliminates the need for long division. It is used to divide polynomials by linear factors of the form (x - a) or (x + a). The steps involved in synthetic division are:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Perform a series of multiplications and additions to find the quotient and remainder.

Step-by-Step Solution

To solve this problem, you need to follow these steps:

Write down the coefficients of the polynomial.
Write down the root of the linear factor.
Perform a series of multiplications and additions to find the quotient and remainder.

Multiplication and Addition

To perform the multiplication and addition, you need to follow these steps:

Multiply the root of the linear factor by the first coefficient.
Add the result to the second coefficient.
Multiply the root of the linear factor by the result.
Add the result to the third coefficient.

Quotient and Remainder

The quotient is the result of the division, and the remainder is the result of the multiplication and addition.

Quotient

The quotient is:

6x - 43

Remainder

The remainder is:

306

Final Answer

The final answer is:

7 \longdiv {6} -1 \quad 5

This option correctly expresses the division problem in synthetic division form.